Dark Enlightenment

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https://en.wikipedia.org/wiki/Dark_Enlightenment

The Dark Enlightenment, also called the Neo-Reactionary movement (abbreviated to NRx), is an anti-democratic, anti-egalitarian, and reactionary philosophical and political movement. It can be understood as a reaction against values and ideologies associated with Enlightenment,advocating for a return to traditional societal constructs and forms of government, such as absolute monarchism and cameralism. The movement promotes the establishment of authoritarian capitalist city-states that compete for citizens. Neoreactionaries refer to contemporary liberal society and its institutions as "the Cathedral", associating them with the Puritan church, and their goals of egalitarianism and democracy as "the Synopsis". They say that the Cathedral influences public discourse to promote progressivism and political correctness which they view as a threat to Western civilization. Additionally, the movement advocates for scientific racism, a view which they say is suppressed by the Cathedral.

Curtis Yarvin began constructing the basis of the ideology in the late 2000s, drawing upon libertarianism and Austrian economics along with thinkers such as Hans-Hermann Hoppe and Thomas Carlyle. Nick Land elaborated upon Yarvin's ideas and coined the term "Dark Enlightenment", applying it to his accelerationism as a means to achieve a technological singularity. The movement has also received contributions from prominent figures, such as venture capitalist Peter Thiel. Despite criticism, the movement has gained traction with parts of Silicon Valley, as well as with several political figures associated with United States President Donald Trump, including political strategist Steve Bannon, Vice President JD Vance, and Michael Anton.

The Dark Enlightenment has been described as part of the alt-right, as its theoretical branch, and as neo-fascist. It has been described as the most significant political theory within the alt-right, as "key to understanding" the alt-right political ideology, and as providing a philosophical basis for considerable amounts of alt-right political activity. University of Chichester professor Benjamin Noys described it as "an acceleration of capitalism to a fascist point". Nick Land disputes the similarity between his ideas and fascism, saying that "Fascism is a mass anti-capitalist movement", whereas he prefers that "capitalist corporate power should become the organizing force in society". Historians Angela Dimitrakaki and Harry Weeks link the Dark Enlightenment to neofascism via Land's "capitalist eschatology", which they argue is grounded in the supremacist theories of fascism. Neoreactionary ideas have also been described as "feudalist", or "techno-feudalist".

History

Curtis Yarvin is one of the founders of the movement.

Neo-reactionaries are an informal community of bloggers and political theorists who have been active since the 2000s. Steve Sailer and Hans-Hermann Hoppe are contemporary forerunners of the ideology, which is also heavily influenced by the political thought of Thomas Hobbes, Thomas Carlyle, and Julius Evola. In 2007 and 2008, software engineer Curtis Yarvin, writing under the pen name Mencius Moldbug, articulated what would develop into Dark Enlightenment thinking. Yarvin's theories were elaborated and expanded by philosopher Nick Land, who first coined the term "Dark Enlightenment" in his essay of the same name.

By mid-2017, NRx had moved to forums such as the Social Matter online forum, the Hestia Society, and Thermidor Magazine. In 2021, Yarvin appeared on Fox News' Tucker Carlson Today, where he discussed the United States' withdrawal from Afghanistan and his concept of the 'Cathedral', which he says is the current aggregation of political power and influential institutions that is controlling the country. Emerson Brooking, an expert in online extremism, said that "Yarvin escaped the fringe blogosphere because he wrapped deeply anti-American, totalitarian ideas in the language of U.S. start-up culture."

Influence in government

Several prominent Silicon Valley investors and Republican politicians have been associated with the philosophy. Steve Bannon has read and admired Yarvin's work, and there have been allegations that he has communicated with Yarvin which Yarvin has denied. Bannon would later consider Yarvin an enemy, which Yarvin did not reciprocate. Michael Anton, the State Department Director of Policy Planning during Trump's second presidency, has also discussed Yarvin's ideas, and Yarvin has claimed to have given staffing recommendations to him. In January 2025, Yarvin attended a Trump inaugural gala in Washington; Politico reported he was "an informal guest of honor" due to his "outsize influence over the Trumpian right." Marc Andreessen has quoted Yarvin and referred to him as a "friend", also investing in his startup Tlon and urging people to read him.

According to historian of conservatism Joshua Tait, "Moldbug's relationship with the investor-entrepreneur Thiel is his most important connection." Max Chafkin described Yarvin as the "house political philosopher" for Thiel's circle of influence (or "Thielverse"), including people such as Blake Masters, and Yarvin has referred to Thiel as "fully enlightened". Vanity Fair noted that both have been influential in the New Right and the National Conservatism Conference. Thiel had also invested in Yarvin's Tlon.

U.S. Vice President JD Vance has cited Yarvin as an influence and has connections to Thiel. Prior to his election to the Vice Presidency, JD Vance cited in his 2022 Senate Campaign Yarvin's "strongman plan to 'retire all government employees,' which goes by the mnemonic 'RAGE.'" In a 2021 interview, "Vance said Trump should 'fire every single midlevel bureaucrat, every civil servant in the administrative state, and replace them with our people. And when the courts stop you, stand before the country and say, 'The chief justice has made his ruling. Now let him enforce it.'" Yarvin has praised Vance, stating "in almost every way, JD is perfect", but also considered his relationship with Vance overstated by the media, as they've rarely communicated. He also praised Trump for breaking from Republican practices of trying to "play ball and help the system work" and instead "trying to move all of the levers of this machine that he can move", though also stating "what he’s doing is not at all what I would do with an opportunity like this. But I think that what I would do is probably not possible."

It has been suggested that the Department of Government Efficiency, or DOGE, bears resemblance to RAGE, as advocated for by Yarvin. Land, when asked by the Financial Times if he approved of DOGE, said "the answer is definitely yes", having also endorsed Steve Bannon's goal of "deconstruction of the administrative state". In a report by The Washington Post, two DOGE advisors described Yarvin as an "intellectual beacon" for the department, with one saying, "It's an open secret that everyone in policymaking roles has read Yarvin." The report said that Yarvin, initially approving of the Trump administration, had become critical of DOGE. He cited its handling of the National Science Foundation and National Institutes of Health, stating "Instead of fighting against these people because they’re an enemy class who votes for the Democrats, you [should be] saying, 'Oooh, we have cookies for you.'" However, Tait said that Yarvin bears some responsibility for DOGE, saying, "It would have been created, probably, regardless. But he spent a good chunk of time creating a justifying framework for it." Political philosopher Danielle Allen said that DOGE is clearly based on Yarvin's work, and the outcome was the natural result of the shortcomings in Yarvin's views. CNN argues that Thiel, Andreessen, Vance and Anton do not deny that they are listening to Yarvin; however, they indicated that they do not accept all of Yarvin's theories:

An advisor to Vance denied the vice president has a close relationship with Yarvin, saying the two have met 'like once.' Thiel, who did not respond to a request for comment, told The Atlantic in 2023 he didn't think Yarvin's ideas would 'work' but found him to be an 'interesting and powerful' historian. And earlier this year [2025], Andreessen, who also did not respond to a request for comment, posted on X that one can read 'Yarvin without becoming a monarchist.'

Beliefs

Opposition to democracy

Central to neoreactionarism's ideas is a belief in freedom's incompatibility with democracy, with Land having stated "Democracy tends to fascism". Yarvin and Land drew inspiration from libertarians such as Thiel, particularly his statement "I no longer believe that freedom and democracy are compatible" in a Cato Unbound essay. Yuk Hui additionally notes Thiel's contribution to the 2004 conference “Politics and Apocalypse” in which he argued that the U.S. needed a new political theory in the face of 9/11, which marked the failure of the Enlightenment, and that democracy and equality had made the West vulnerable. However, when asked by The Atlantic about Yarvin, Thiel opined that trying to radically alter the current U.S. government was unrealistic. He also suggested that Yarvin's methods would lead to Xi's China or Putin's Russia. Hui notes that neoreactionaries consider the Enlightenment values of democracy and equality to be degenerative and limiting, respectively. Tait considers Yarvin to have "a complex relationship" with Enlightenment values, as he adopts a secular and rationalist view of reality while rejecting its key political ideals of equality and democracy. Sergio C. Fanjul contrasts the movement's far-right critique of the Enlightenment with the Frankfurt School's critique of the Enlightenment as a Eurocentric prelude to colonialism and war.

Yarvin told Vanity Fair: "The fundamental premise of liberalism is that there is this inexorable march toward progress. I disagree with that premise." A 2016 article in New York magazine notes that "Neoreaction has a number of different strains, but perhaps the most important is a form of post-libertarian futurism that, realizing that libertarians aren't likely to win any elections, argues against democracy in favor of authoritarian forms of government." Journalist Andrew Sullivan writes that neoreaction's pessimistic appraisal of democracy dismisses many advances that have been made and that global manufacturing patterns also limit the economic independence that sovereign states can have from one another.

Support for authoritarianism

Yarvin supports authoritarianism on right-libertarian grounds, saying that the division of political sovereignty expands the scope of the state, whereas strong governments with clear hierarchies remain minimal and narrowly focused. Yarvin's "A Formalist Manifesto" advocates for a form of "neocameralism" in which small, authoritarian "gov-corps" coexist and compete with each other, an idea anticipated by Hans-Hermann Hoppe. Academic Jonathan Ratcliffe describes the model as "a network of hyper-capitalist city states ruled by authoritarian CEO monarchs." Yarvin claims freedom under the system, known as the "Patchwork", would be guaranteed by the ability to "vote with your feet", whereby residents could leave for another gov-corp if they felt it would provide a higher quality of life, thus forcing competition. Land reiterates this with the political idea "No Voice, Free Exit", taken from Albert Hirschman's Exit, Voice, and Loyalty model in which voice is democratic and exit is departure to another society:

"If gov-corp doesn’t deliver acceptable value for its taxes (sovereign rent), [citizens] can notify its customer service function, and if necessary take their custom elsewhere. Gov-corp would concentrate upon running an efficient, attractive, vital, clean, and secure country, of a kind that is able to draw customers."

Yarvin has advocated for a "dictator-president" or "national CEO". He has described himself as a royalist, monarchist, and Jacobite; and has praised cameralism, Frederick the Great, and Thomas Carlyle. He is also influenced by Austrian economics, particularly Hoppe, Ludwig von Mises, Murray Rothbard, and Friedrich Hayek. Ava Kofman credits Hoppe's Democracy: The God That Failed with pushing Yarvin away from standard libertarian thought, with authoritarianism scholar Julian Waller saying "it's not copy-and-pasted, but it is such a direct influence that it's kind of obscene". Patrick Gamez notes that Land is "simply catching up to Murray Rothbard, Hans-Hermann Hoppe, Peter Brimelow, and assorted other radically right-wing libertarians and anarcho-capitalists, committed to 'cracking up' the democratic nation-state in favor of an 'ethno-economy.'"

Yarvin admires Chinese leader Deng Xiaoping for his pragmatic and market-oriented authoritarianism, and the city-state of Singapore as an example of a successful authoritarian regime. He sees the US as soft on crime, dominated by economic and democratic delusions. He additionally cites Dubai and Hong Kong as providing a high quality of life without democracy, stating "as Dubai in particular shows, a government (like any corporation) can deliver excellent customer service without either owning or being owned by its customers."

Andy Beckett stated that NRx supporters "believe in the replacement of modern nation-states, democracy and government bureaucracies by authoritarian city states, which on neoreaction blogs sound as much like idealised medieval kingdoms as they do modern enclaves such as Singapore." Ana Teixeira Pinto describes the political ideology of the gov-corp model as a form of classical libertarianism, stating "they do not want to limit the power of the state, they want to privatise it." According to criminal justice professor George Michael, neoreaction seeks to perform a "hard reset" or "reboot" on democracy rather than gradual reform. Neoreactionary ideas have also been referred to as "feudalist" and "techno-feudalist". Yarvin's proposals are not fully detailed beyond philosophy and general principles, and the economic ability to leave and the willingness of other locations to accept immigrants are not generally considered. Andrew Jones criticized his arguments as "vaguely defined and often factually incorrect".

The process of instituting authoritarianism

Yarvin describes his proposals as a modern version of monarchy and advocates for an American monarch dissolving elite academic institutions and media outlets within the first few months of their reign, stating "if Americans want to change their government, they're going to have to get over their dictator phobia." Time notes that Yarvin's proposal for a "Butterfly Revolution" envisions an internal coup to replace democracy with a privatized executive authority, which includes his RAGE proposal to "retire all government employees" in favor of loyalists. While conceding that it may not be possible, he stated that, were he in Trump's position, he would take executive control of government institutions such as the Federal Reserve, keeping those "that have a very clear role and are not politicized in any way" while disposing of others such as the State Department. He advocates constitutionally challenging laws such as impoundment control, birthright citizenship, and Marbury v. Madison, potentially defying the courts if it were necessary and "unifying". However, he also stated "if you're doing that in a situation where the vibe is like, 'This is going to be the first shot in the civil war between red America and blue America' ... I think it’s bad", considering Trump and America "unready for that level of change".

He suggested in a January 2025 New York Times interview that there was historical precedent to support his reasoning, asserting that in his first inaugural address, Franklin Delano Roosevelt "essentially says, Hey, Congress, give me absolute power, or I'll take it anyway. So did FDR actually take that level of power? Yeah, he did." The interviewer, David Marchese, remarked that "Yarvin relies on what those sympathetic to his views might see as a helpful serving of historical references — and what others see as a highly distorting mix of gross oversimplification, cherry-picking and personal interpretation presented as fact." Scholars have described Yarvin's arguments as misrepresenting the historical record, and said that the historical autocracies he praises were considered deeply oppressive by their subjects.

The Cathedral

Neoreactionaries refer to contemporary liberal society and institutions which they oppose as the "Cathedral", considering them the descendant of the Puritan church, and their goals of egalitarianism and democracy as "the Synopsis". According to them, the Cathedral influences public discourse to promote progressivism and political correctness, and its adoption of liberal humanism is the primary reason for an alleged decline of Western civilization. A neoreactionary online dictionary defines the Cathedral as "the self-organizing consensus of Progressives and Progressive ideology represented by the universities, the media, and the civil service", with an agenda that includes "women’s suffrage, prohibition, abolition, federal income tax, democratic election of senators, labor laws, desegregation, popularization of drugs, destruction of traditional sexual norms, ethnic studies courses in colleges, decolonization, and gay marriage." Yarvin views it as an oligarchy of educated elites competing for status, and has accused Ivy League schools, The New York Times, and Hollywood of being members.

Land and others argue that enforcement of political correctness by these institutions means that they are a religious entity, hence the term 'Cathedral'. Yarvin, described by El País as a former progressive, describes these institutions as a "twentieth-century version of the established church", with the educational system as a method for indoctrinating people into the Cathedral, enforcing compliance with progressive ideology and preventing them from thinking for themselves. Yarvin defines a church as "an organization or movement which tells people how to think", and includes schools as churches.

The concept of the Cathedral has been described as "fundamental to the alt-right's understanding of the humanities". Academic Andrew Woods describes the Cathedral as one of two central ideas that enable the alt-right to dismiss criticism, the other being cultural Marxism. He writes that both ideas function to pre-emptively neutralize attempts at refutation, and that they are especially used to delegitimize critical theory. The Cathedral allegedly "seeks to delude the American public" while amassing power and influence, and critical theory is portrayed as the ideological justification for the pursuit of power. Progressive thought is seen as a disguise for power-seeking, and Woods says that Yarvin takes advantage of the inability to prove the unconscious desires of others to argue that "everyone's primary motivation in life is their craving for greater power." El País compared the concept to QAnon and its claims of a deep state.

Race

Neoreactionaries endorse scientific racism, a pseudoscientific view which they refer to as "human biodiversity". Land coined the term "hyperracism" to refer to his views on race; he believes that socioeconomic status is "a strong proxy for IQ" rather than race specifically (though he acknowledges a correlation between race and socioeconomic status), and that meritocracy, particularly space colonization, will "function as a highly-selective genetic filter" that propagates mostly (but not strictly) Whites and Asians. Roger Burrows, writing for The Sociological Review, stated "In Land's schema, the consumers ‘exiting’ from competing gov-corps quickly form themselves into, often racially based, microstates. Capitalist deterritorialization combines with ongoing genetic separation between global elites and the rest of the population resulting in complex new forms of ‘Human Bio-diversity’. He described Land's views as eugenicist and compared them to those of The Bell Curve.

According to Land, the concepts of hate speech and hate crimes are simply methods to suppress ideas that contradict the Cathedral's dogma. He says that statements described as "hate speech" are not related to hatred but are simply a type of defiance of the Cathedral's religious orthodoxy. The suppression is carried out by the "Media-Academic Complex" because the ideas are seen as reflecting a "heretical intention". Yarvin has stated, "Although I am not a white nationalist, I am not exactly allergic to the stuff", believing it to simply be an ineffective tool for "the very real problems about which it complains." Yarvin has endorsed arguments for black racial inferiority and says they are being suppressed by the Cathedral. He has said that some races are more suited to slavery than others and has been described as a modern-day supporter of slavery, a description he disputes.

Accelerationism

One of Land's goals with neoreactionarism is to drive accelerationism. Roger Burrows stated of Land's interpretation of Yarvin, "The Dark Enlightenment itself might be best thought of as the application of Land’s accelerationist framework to Moldbug’s neocameralism." Land views democratic and egalitarian policies as only slowing down acceleration and a technocapital singularity, stating "Beside the speed machine, or industrial capitalism, there is an ever more perfectly weighted decelerator ... comically, the fabrication of this braking mechanism is proclaimed as progress. It is the Great Work of the Left." Vincent Le states "If Land is attracted to Moldbug’s political system, it is because a neocameralist state would be free to pursue long-term technological innovation without the democratic politician’s need to appease short-sighted public opinion to be re-elected every few years."

Vox attributed such views to Land living in China's "techno-authoritarian political system" and his admiration for Deng Xiaoping and Singapore's Lee Kuan Yew. Land has referred to Lee as an "autocratic enabler of freedom", and Yarvin has also praised Lee. Yuk Hui considers sinofuturism to be a model for the movement's pursuit of technological progress which results from a perceived decline of the West. According to Hui, political fatigue leads people such as Land to look towards Asian cities such as Shanghai, Hong Kong, and Singapore as examples of "depoliticized techno-commercial utopia". China is viewed as smoothly importing Western science and technology while Western innovation is constantly limited by the progressivism of the Cathedral. Hui considers this to be "simply a detached observation of these places that projects onto them a common will to sacrifice politics for productivity". Land has advocated for accelerationists to support the neoreactionary movement, though many have distanced themselves from him in response to his views on race.

Formalism

In the inaugural article published on Unqualified Reservations in 2007, entitled "A Formalist Manifesto", Yarvin used the term "formalism" for his ideas, advocating for the formal recognition of the realities of existing power by aligning property rights with current political power as a solution to violence. Courtney Hodrick, writing for Telos, stated "in his view, all politics are individual property relationships and the social contract is an agreement between citizen-consumers and governor-owners. Your consent to an agreement such as 'I won’t kill anyone on the street,' he explains, is 'just your agreement with whoever owns the street.' This agreement means that the owner of the street may use violence to enforce this agreement, just as individuals may use violence to defend their own property. His concern ... is deciding who has the monopoly on the legitimate use of violence. But rather than concern himself with justifying legitimacy politically or metaphysically, Moldbug calls for a naturalization of existing property relations." Yarvin describes the U.S. as "an big [sic] old company that holds a huge pile of assets, has no clear idea of what it’s trying to do with them, and is thrashing around like a ten-gallon shark in a five-gallon bucket", advocating formalism as a solution:

"To a formalist, the way to fix the US is to dispense with the ancient mystical horseradish, the corporate prayers and war chants, figure out who owns this monstrosity, and let them decide what in the heck they are going to do with it. I don't think it's too crazy to say that all options—including restructuring and liquidation—should be on the table."

Yarvin rejects democracy as "ineffective and destructive" and attributes the successes of the post-World War II democratic system to its actually being "a mediocre implementation of formalism". He describes democratic politics as "a sort of symbolic violence, like deciding who wins the battle by how many troops they brought". Rejecting pacifism for what he perceives as a tendency to advocate for the rectification of injustices instead of seeking an end to armed conflict, Yarvin promotes the adoption of classical approaches to international law and the idea of "formalising the military status quo" as the most direct path to peace. He identifies the form of pacifism which prioritises "righteousness" instead of peace with the Calvinist doctrine of providence, and "ultracalvinism" as the ideological/theological basis for contemporary American interventionism.

Relation to other movements

Seasteading

Prominent figures in the neoreactionary movement have connections to seasteading, the creation of sovereign city-states in international waters, which has been characterized as a way to execute the movement's ideas. Yarvin has connections to Patri Friedman, founder of The Seasteading Institute and grandson of Milton Friedman, and Thiel was once its main investor. Thiel has also advocated the use of cyberspace, outer space, and the oceans to outstrip traditional politics via capitalism in order to realize libertarianism. Land has quoted Friedman in stating that "free exit is so important that…it [is] the only Universal Human Right".

The Network State

See also: City of Starbase Incorporation election and Free State Project

Balaji Srinivasan's proposal of the Network State, a plan for technology executives and investors to remove themselves from democracy and create their own sovereign states, has been compared to Yarvin's ideas. Journalists have described Srinivasan as a leader of the neoreactionary movement and a friend of Yarvin. Srinivasan had also messaged Yarvin suggesting potentially using the Dark Enlightenment audience to dox reporters. Comparisons have also been made to Galt's Gulch from Atlas Shrugged and Donald Trump's proposed "Freedom Cities". Supporters include Marc Andreessen, Garry Tan, Peter Thiel, Michael Moritz, Patrick Collison, Patri Friedman, Roger Ver, Naval Ravikant, Joe Lonsdale, Bryan Johnson, the Winklevoss twins. Sam Bankman-Fried, Sam Altman, Shervin Pishevar, Brian Armstrong, and Vitalik Buterin. Proposed cities alleged to be examples of the Network State include California Forever, Praxis, Telosa, Neom, Liberland, and Elon Musk's Starbase City. Established cities alleged to be part of the Network state include Próspera in Honduras and Itana in Nigeria. Other locations of interest include Greenland, French Polynesia, Palau, South Asia, Ghana, the Marshall Islands, Panama, the Bahamas, Montenegro, Costa Rica, and Rhode Island.

Cryptocurrency and Web3 are central components of the project. Its legal framework also involves special economic zones, and foreign investors have used Investor–state dispute settlement (ISDS) in the case of Próspera. The movement has been compared to Trumpism, with common ideologies including a belief in Strauss–Howe generational theory and hostility to left-wing politics, the news media and the administrative state. Critics have described these projects as a form of neocolonialism, corporate monarchy, or white saviorism. The Highland Rim Project, located in Tennessee and Kentucky, is a Christian nationalist community influenced by the Network State and was proposed by New Founding, a Christian venture capital firm that received funding from Andreessen and is connected to the Network State venture capital firm Pronomos Capital. The Guardian has noted the community's ties to far-right groups and white nationalism.

Surveillance capitalism

See also: Surveillance capitalism

Mother Jones cites Clearview AI and its founder Hoan Ton-That (who were in connection with Thiel and Yarvin) as an example of the Dark Enlightenment or neoreactionary thinking's influence on the development of surveillance technology. A 2025 anonymous letter of a group of self-described former followers of the neoreactionary movement warned that the movement advocated for "techno-monarchism" in which its ruler would use "data systems, artificial intelligence, and advanced algorithms to manage the state, monitor citizens, and implement policies". It further warned that Musk, in the context of his actions at the Department of Government Efficiency, was working "for his own power and the broader neo-reactionary agenda." Yarvin has outlined a vision for San Francisco where public safety would be enforced by constant monitoring of residents and visitors via RFID, genotyping, iris scanning, security cameras, and transportation which would track its location and passengers, reporting all of it to the authorities. The New Republic described the proposed surveillance system as "Orwellian".

Alt-right

The Dark Enlightenment has been described by journalists and commentators as part of the alt-right, specifically as its theoretical branch. Journalist and pundit James Kirchick states that "although neo-reactionary thinkers disdain the masses and claim to despise populism and people more generally, what ties them to the rest of the alt-right is their unapologetically racist element, their shared misanthropy and their resentment of mismanagement by the ruling elites".

Scholar Andrew Jones wrote in 2019 that the Dark Enlightenment is the most significant political theory within the alt-right, and that it is "key to understanding" the alt-right political ideology. "The use of affect theory, postmodern critiques of modernity, and a fixation on critiquing regimes of truth", Jones remarked, "are fundamental to NeoReaction (NRx) and what separates it from other Far-Right theory". Moreover, Jones argues that Dark Enlightenment's fixation on aesthetics, history, and philosophy, as opposed to the traditional empirical approach, distinguishes it from related far-right ideologies. Historian Joe Mulhall, writing for The Guardian, described Land as "propagating very far-right ideas." Despite neoreaction's limited online audience, Mulhall considers the ideology to have "acted as both a tributary into the alt-right and as a key constituent part [of the alt-right]." Journalist Park MacDougald described neoreactionarism as providing a philosophical basis for considerable amounts of alt-right political activity.

The term "accelerationism", originally referring to Land's technocapitalist ideas, has been re-interpreted by some into the use of racial conflict to cause societal collapse and the building of white ethnostates, which has been linked to several white nationalist terrorist attacks such as the 2019 Christchurch mosque massacres. Vox described Land's shift towards neoreactionarism, along with neoreactionarism crossing paths with the alt-right as another fringe right wing internet movement, as the likely connection point between far-right racial accelerationism and the otherwise unrelated technocapitalist term. They cited a 2018 Southern Poverty Law Center investigation which found users on the neo-Nazi blog The Right Stuff who cited neoreactionarism as an influence. Land himself has called the neoreactionary movement "a prophetic warning about the rise of the Alt-Right".

Fascism

Journalists and academics have described the Dark Enlightenment as neo-fascist. University of Chichester professor Benjamin Noys described it as "an acceleration of capitalism to a fascist point". Nick Land disputes the similarity between his ideas and fascism, saying that "Fascism is a mass anti-capitalist movement", whereas he prefers that "capitalist corporate power should become the organizing force in society". Historians Angela Dimitrakaki and Harry Weeks tie the Dark Enlightenment to neofascism via Land's "capitalist eschatology" which they describe as supported by the supremacist theories of fascism. Dimitrakaki and Weeks say that Land's Dark Enlightenment was "infusing theoretical jargon into Yarvin/Moldbug's blog 'Unqualified Reservations'".

In The Sociological Review, Roger Burrows examined neoreaction's core tenets and described the ideology as "hyper-neoliberal, technologically deterministic, anti-democratic, anti-egalitarian, pro-eugenicist, racist and, likely, fascist", and describes the entire accelerationist framework as a faulty attempt at "mainstreaming ... misogynist, racist and fascist discourses". He criticizes neoreaction's racial principles and its brazen "disavowal of any discourses" advocating for socio-economic equality and, accordingly, considers it a "eugenic philosophy" in favor of what Nick Land deems "hyper-racism". Graham B. Slater wrote that neoreaction "aim[s] to solve the problems purportedly created by democracy through what ultimately amount to neo-fascist solutions."

Land himself became interested in the Atomwaffen-affiliated theistic Satanist organization Order of Nine Angles (ONA) which adheres to the ideology of Neo-Nazi terrorist accelerationism, describing the ONA's works as "highly-recommended" in a blog post. In the contemporary art world, art historian Sven Lütticken says that the popularity of Land's concepts has made certain art centers in New York and London hospitable to trendy fascism.

Authoritarian capitalism

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Authoritarian_capitalism

Authoritarian capitalism, or illiberal capitalism, is an economic system in which a liberal capitalist market economy exists alongside an authoritarian government. It overlaps significantly with state capitalism, a system in which the state undertakes commercial activities. However, it is distinct in its combination of private property and the functioning of market forces with restrictions on dissent, a complete lack of freedom of speech or significant limits on it, and either an electoral system with a single dominant political party or a lack of elections.

Countries commonly referred to as being authoritarian capitalist states include China since the reform and opening up; Russia, under Vladimir Putin; Chile, under Augusto Pinochet; Indonesia, under Suharto; Peru under Alberto Fujimori and Singapore, under Lee Kuan Yew. Additionally, the term is often applied to military dictatorships that received support from the United States during the Cold War era.

Political scientists disagree on the long-run sustainability of authoritarian capitalism, with arguments both for and against the long-term viability of political repression alongside a capitalist free-market economic system.

History

Early development

As a political economic model, authoritarian capitalism is not a recent phenomenon. Throughout history, examples of authoritarian capitalism include Manuel Estrada Cabrera's and Jorge Ubico's respective reigns in Guatemala, Augusto Pinochet's reign in Chile, Suharto's New Order in Indonesia and the People's Action Party's early administration in Singapore. During World War I, the ideological divide between authoritarian and liberal regimes was significantly less pronounced as both were aligned to capitalist economic models. Moreover, the Axis powers of World War II have been described as possessing totalitarian capitalist economic systems, acting as examples of the early developments of authoritarian capitalism.

From the end of World War II, various authoritarian capitalism regimes emerged, developed and transitioned into a liberal capitalist model through East Asia, Southern Europe and Latin America. It has been argued that the change of these early regimes was predominately due to the dominance of liberal capitalist countries such as the United States as opposed to a natural transition, suggesting that modern authoritarian capitalist regimes may further develop the system.

Recent prominence

China's Xi Jinping and Russia's Vladimir Putin, both prominent recent figures for authoritarian capitalism

While having been a relatively unknown system due to the failure of authoritarianism within the First World during the Cold War, with the transition of authoritarian countries such as China and Russia to capitalist economic models, authoritarian capitalism has recently risen to prominence. While it was initially thought that changing to a capitalist model would lead to the formation of a liberal democracy within authoritarian countries, the continued persistence of an authoritarian capitalist models has led to this view decreasing in popularity. Furthermore, some have argued that by using capitalist economic models authoritarian governments have improved the stability of their regimes through improving the quality of life of their citizenship. Highlighting this appeal, Robert Kagan stated: "There's no question that China is an attractive model for autocrats who would like to be able to pursue economic growth without losing control of the levers of power".

Moreover, authoritarian capitalist regimes have experienced notable growth in their economic production, with the International Monetary Fund stating that authoritarian capitalist countries experienced an average 6.28% GDP growth rate compared to the 2.62% of liberal capitalist countries. In addition, many have argued the inability of liberal capitalism, with the 2008 financial crisis and the slow response of the United States government, to quickly respond to crisis compared to more authoritarian systems has been bought into prominence. In fact, many argue that authoritarian capitalism and liberal capitalism have or will compete on the global stage.

According to political economist Radhika Desai, a Marxist scholar, certain factions of the capitalist class in the West, especially those who support the politics of Donald Trump in the US and Boris Johnson in the UK, favor a more authoritarian capitalism, often embracing protectionism, xenophobia, racism and misogyny as being complementary to economic neoliberalism.

State capitalism

Main article: State capitalism

Within countries that practice authoritarian capitalism, state capitalism is generally also present to some extent and vice versa. As such, there is a widespread confusion between the terms with them at times being treated as synonymous by individuals such as former Australian Prime Minister Kevin Rudd.

Overlap

Authoritarian governments often seek to establish control within their borders and, as such, will use state-owned corporations; therefore, state capitalism will emerge to some extent within countries that practice authoritarian capitalism, manifesting from the ruling authority's desire to exercise control. The prominent use of state-owned corporations and sovereign wealth funds within authoritarian capitalist regimes demonstrates such a tendency, with Russia decreasing its private ownership of oil from 90% to 50% while transitioning to a more authoritarian model under the leadership of Vladimir Putin.

It has also been noted by individuals such as Richard W. Carney that authoritarian regimes have a strong tendency to use their economies to increase their influence, heavily investing in their economies through state-owned enterprises. Carney describes the intervention of authoritarian states occurring through means he describes as extra-shareholder tactics, including regulations, government contracts, and protectionist policies alongside the state engaging in shareholder activism. Moreover, he focuses on the use of state-owned funds to engage in takeovers of key assets in other countries, such as Khazanah Nasional's takeover of Parkway Pantai in 2010.

Differences

There remains a fundamental difference with state capitalism being a system in which government owned entities engage in for-profit activities while authoritarian capitalism is a system where an authoritarian regime co-exists with, or at least attempts to adopt aspects of, a market economy, highlighted in countries such as Hungary by the Transnational Institute.

Examples

China

Further information: Economy of China

It is generally agreed that China is an authoritarian regime, with the Fraser Institute ranking them 136th for personal freedom and the Human Right Watch's 2018 report describing a "broad and sustained offensive on human rights" within China due to the treatment of activists, restrictions to freedom of information, political expression, religious freedom and minority rights as their core reasons. Michael Witt argues that China broadly displays capitalist traits with a significant number of companies either being private or shared between private and public owners alongside a strong entrepreneurial presence despite a continued predominance of indirect state control.

Reporter Joseph Kurlantzick and political scientist Yuen Yuen Ang state that China is unable to fully use the entrepreneurial elements needed to drive future growth if it maintains authoritarian control. As Ang writes in Foreign Affairs, "[t]o achieve this kind of growth, the government must release and channel the immense creative potential of civil society, which would necessitate greater freedom of expression, more public participation, and less state intervention". As stated by Joseph Kurlantzick, "China's growth 'model' has shown impressive resilience in recent years", with an ability to rapidly respond to crises, confidence around economic success and growing soft power being used to explain it.

Russia

Further information: Economy of Russia

Putin and Boris Yeltsin, both prominent figures of the development of authoritarian capitalism in Russia

Azar Gat describes Russia, along with China, as a prominent example of a modern authoritarian capitalist nation, describing the country as becoming increasingly authoritarian while maintaining a predominately capitalist economic model. Aaron Friedberg simplifies the evolution of the Russian model in the following statement: "The Russian system has also evolved from communist totalitarianism to a form of nationalist authoritarian capitalism that appears for the moment at least to be relatively stable". Friedberg also describes the 1996 presidential election as the point where authoritarian capitalism began forming within Russia, depicting an increasingly powerful majority party backed by media controlled by oligarchies and led by Boris Yeltsin and later Vladimir Putin. From 1999 under Putin, Friedberg describes the Russian regime as solidifying its power through re-obtaining state control of natural resources, obtaining control of media, and limiting dissidence through measures such as restricting non-governmental organization operations.

Saudi Arabia

Saudi Arabia is considered by Freedom House as an authoritarian country, receiving a civil liberty score of 7/100; this is due to the fact that the Saudi absolute monarchy prevents any political rights, and the government extremely controls freedom of expression and religion. The Saudi economy has been opening the market regularly even though it has nationalized some state-owned companies.

Singapore

Further information: Economy of Singapore

Lee Kuan Yew, a major figure in the development of Singapore's economic model

Singapore is considered by agencies such as the Human Rights Watch as a highly repressive regime. They describe a lack of freedom of speech, capital punishment, detention without trial, and sexual freedom as causing the country to run contrary to international human rights. Moreover, the country under the rule of Lee Kuan Yew has been described as embracing the core aspects of capitalism, with the Fraser Institute ranking it second for economic freedom in 2016, creating a state of authoritarian capitalism. However, there is contention around the continued viability of Singapore's economics success which has increased its GDP per capita from US$427.88 in 1960 to US$57,714.3 in 2017. Some economists argue that Singapore has severely restricted its ability to obtain future growth through the repression of individual freedom of expression and thought. Regardless of this, Singapore is considered as an exception in regards to its stability, with Daniel W. Drezner stating that "with the exception of Singapore, this model has never worked over the long run".

Contention

Authoritarian capitalism is a political-economic model that has faced a variety of criticism. Some experts state that the authoritarian capitalist model is unstable and will eventually transition into that of liberal capitalism, with Daniel W. Drezner stating: "The conventional wisdom in comparative politics is that as societies get richer ... they also start demanding more political accountability." In opposition, others argue that the increased wealth of capitalist regimes allows authoritarian regimes to more adequately utilize technology to assist in maintaining their regimes.

Criticism

Daniel W. Drezner, writing for Foreign Policy magazine, argues that when societies get richer, their citizens start demanding more political accountability and democracy. Therefore, capitalist economic policies that successfully promote economic growth will be inherently detrimental to the continuation of an authoritarian regime. Individuals will increasingly seek to reduce restrictions upon their human rights as their quality of life and access to communication resources increase, so a successful economy will inevitably lead to citizens revolting against authoritarian governments. An appropriate example of this is the Imperial State of Iran during the reign of the Shah Mohammad Reza Pahlavi which was an authoritarian state capitalist system that enjoyed incredible growth but which nonetheless led to revolution.

Yuen Yuen Ang, writing in Foreign Affairs, argues that the restrictions to freedom of expression found in authoritarian regimes are harmful to the ability of citizens to innovate and engage in entrepreneurship, leading to a reduction in the economic growth of the country. John Lee, Michael Witt and Gordon Redding claim that authoritarian capitalist regimes primarily obtain their legitimacy through their ability to deliver economic growth, and therefore this inherent restriction upon economic growth would eventually lead to the collapse of the regime.

Authoritarian capitalist regimes are viewed as having to face civil disobedience towards their authoritarian characteristics, exhibited by countries such as China experiencing 87,000 instances of mass unrest in 2005. Some critical scholars have argued that "authoritarian capitalism" is a redundancy, as capitalism necessarily involves an authoritarian relationship at the micro-economic level (i.e., in the workplace).

Defense

John Lee and Brahma Chellaney have argued that authoritarian capitalism is a potential competitor with liberal capitalism, with the recent success of authoritarian capitalist regimes such as China being used as the core of their argument. Chellaney has further stated that through using elements of capitalism, regimes may more effectively employ modern technologies to suppress dissidence towards government such as the Great Firewall used within China. Niv Horesh also argues that authoritarian capitalist model offered by China is a viable alternative to liberal capitalism, with more effective decision-making processes.

In addition, Niv Horesh holds that capitalist free-market policies lead to an increase in authoritarian policies such as those pursued by Margaret Thatcher. The core of this argument lies in the view that citizens will support whichever regime provides material comforts which increasing economic inequality and automation in liberal capitalist nations undermine. Moreover, challenges to liberal capitalism from an inability to adequately cope with advances of technology have also been raised, summarised in the statement by former Australian Prime Minister Kevin Rudd: "Democracies, like corporations, can now be hacked." Alongside these technological challenges, Michael Witt and Gordon Redding have also pointed to a seeming failure to address structural issues such as gerrymandering. Anders Corr has described the expansion of China as a compelling argument for the success of its authoritarian capitalist regime.

Aaron Friedberg of the Sasakawa Peace Foundation has argued that authoritarian capitalist nations have used an exploitation of the Western world, the reshaping of the international order and exclusion of international actors in an attempt to establish their systems of governance. He has also stated that unlike in the Cold War contemporary authoritarian powers are likely to be driven towards cooperation in their attempts to consolidate their regimes.

Impact on business

In recent years, the Ease of Doing Business index for the authoritarian capitalist states of Hungary and Poland has remained relatively stable, while Singapore continues to lead globally and China has risen sharply. This evidence suggests that authoritarian capitalism can be very business-friendly and attractive for business.

Cubic equation

From Wikipedia, the free encyclopedia

Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis at y = 0). The case shown has two critical points. Here the function is ${\displaystyle \begin{array}{rl}f(x)& =\frac{1}{4}\left(x^3+3x^2-6x-8\right)\\ & =\frac{1}{4}(x-2)(x+1)(x+4)\end{array}}$ and therefore the three real roots are 2, −1 and −4.

In algebra, a cubic equation in one variable is an equation of the form $${\displaystyle a{x}^3+b{x}^2+cx+d=0}$$ in which a is not zero.

The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:

• algebraically: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
• geometrically: using Omar Khayyam's method.
• trigonometrically
• numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.

The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.

History

Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians. Babylonian (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task which is now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century.

In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations (Diophantine equations). Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like T. L. Heath, who translated all of Archimedes's works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two conics, but also discussed the conditions where the roots are 0, 1 or 2.

Graph of the cubic function f(x) = 2x³ − 3x² − 3x + 2 = (x + 1) (2x − 1) (x − 2)

In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x³ + px² + qx = N, 23 of them with p, q ≠ 0, and two of them with q = 0.

In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote:

“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”

In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: x³ + 12x = 6x² + 35. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Muʿādalāt (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the Horner–Ruffini method to numerically approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.

In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x³ + 2x² + 10x = 20. Writing in a sexagesimal numeral system he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/60² + 42/60³ + 33/60⁴ + 4/60⁵ + 40/60⁶), which has a relative error of about 10⁻⁹.

In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x³ + mx = n. In fact, all cubic equations can be reduced to this form if one allows m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it.

Niccolò Fontana Tartaglia

In 1535, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fior, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form x³ + mx = n, for which he had worked out a general method. Fior received questions in the form x³ + mx² = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Later, Tartaglia was persuaded by Gerolamo Cardano (1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution).

Cardano's promise to Tartaglia said that he would not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano from Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income.

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers.

François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète.

Factorization

If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators. Such an equation $${\displaystyle a{x}^3+b{x}^2+cx+d=0,}$$ with integer coefficients, is said to be reducible if the polynomial on the left-hand side is the product of polynomials of lower degrees. By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients.

Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form $${\displaystyle qx-p,}$$ with q and p being coprime integers. The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a, and p must be a divisor of d).

Thus, one root is ${\displaystyle x_1=\frac{p}{q},}$ and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is $${\displaystyle \frac{a}{q}{x}^2+\frac{bq+ap}{q^2}x+\frac{c{q}^2+bpq+a{p}^2}{q^3}.}$$ (The coefficients seem not to be integers, but must be integers if ⁠${\displaystyle p/q}$⁠ is a root.)

Then, the other roots are the roots of this quadratic polynomial and can be found by using the quadratic formula.

Depressed cubic

Cubics of the form $${\displaystyle t^3+pt+q}$$ are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple change of variable to that of a depressed cubic.

Let $${\displaystyle a{x}^3+b{x}^2+cx+d=0}$$ be a cubic equation. The change of variable $${\displaystyle x=t-\frac{b}{3a}}$$ gives a cubic (in t) that has no term in t². In fact, ${\displaystyle x=-\frac{b}{3a}}$ is the inflection point of the original cubic (where the curvature changes sign), so the transformation simply centers the cubic around the inflection point

After dividing by a one gets the depressed cubic equation $${\displaystyle t^3+pt+q=0,}$$ with $${\displaystyle \begin{array}{rl}t=& x+\frac{b}{3a}\\ p=& \frac{3ac-b^2}{3a^2}\\ q=& \frac{2b^3-9abc+27a^{2}d}{27a^3}.\end{array}}$$

The roots ${\displaystyle x_1,x_2,x_3}$ of the original equation are related to the roots ${\displaystyle t_1,t_2,t_3}$ of the depressed equation by the relations $${\displaystyle x_i=t_i-\frac{b}{3a},}$$ for ${\displaystyle i=1,2,3}$.

Discriminant and nature of the roots

The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant.

Discriminant

The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.

If r₁, r₂, r₃ are the three roots (not necessarily distinct nor real) of the cubic ${\displaystyle a{x}^3+b{x}^2+cx+d,}$ then the discriminant is $${\displaystyle a^4(r_1-r_2{)}^2(r_1-r_3{)}^2(r_2-r_3{)}^2.}$$

The discriminant of the depressed cubic ${\displaystyle t^3+pt+q}$ is $${\displaystyle -\left(4p^3+27q^2\right).}$$

The discriminant of the general cubic ${\displaystyle a{x}^3+b{x}^2+cx+d}$ is $${\displaystyle 18abcd-4b^{3}d+b^2{c}^2-4a{c}^3-27a^2{d}^2.}$$ It is the product of ${\displaystyle a^4}$ and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as $${\displaystyle \frac{4(b^2-3ac{)}^3-(2b^3-9abc+27a^{2}d{)}^2}{27a^2}.}$$

It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real, the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants.

To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in r₁, r₂, r₃, and a. The proof then results in the verification of the equality of two polynomials.

Nature of the roots

If the coefficients of a polynomial are real numbers, and its discriminant ${\displaystyle \mathrm{\Delta }}$ is not zero, there are two cases:

• If ${\displaystyle \mathrm{\Delta }>0,}$ the cubic has three distinct real roots
• If ${\displaystyle \mathrm{\Delta }<0,}$ the cubic has one real root and two non-real complex conjugate roots.

This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.

As stated above, if r₁, r₂, r₃ are the three roots of the cubic ${\displaystyle a{x}^3+b{x}^2+cx+d}$, then the discriminant is $${\displaystyle \mathrm{\Delta }=a^4(r_1-r_2{)}^2(r_1-r_3{)}^2(r_2-r_3{)}^2}$$

If the three roots are real and distinct, the discriminant is a product of positive reals, that is ${\displaystyle \mathrm{\Delta }>0.}$

If only one root, say r₁, is real, then r₂ and r₃ are complex conjugates, which implies that r₂ − r₃ is a purely imaginary number, and thus that (r₂ − r₃)² is real and negative. On the other hand, r₁ − r₂ and r₁ − r₃ are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is ${\displaystyle \mathrm{\Delta }<0.}$

Multiple root

If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real.

The discriminant of the depressed cubic ${\displaystyle t^3+pt+q}$ is zero if ${\displaystyle 4p^3+27q^2=0.}$ If p is also zero, then p = q = 0 , and 0 is a triple root of the cubic. If ${\displaystyle 4p^3+27q^2=0,}$ and p ≠ 0 , then the cubic has a simple root $${\displaystyle t_1=\frac{3q}{p}}$$

and a double root $${\displaystyle t_2=t_3=-\frac{3q}{2p}.}$$

In other words, $${\displaystyle t^3+pt+q=\left(t-\frac{3q}{p}\right){\left(t+\frac{3q}{2p}\right)}^2.}$$

This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas.

By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic ${\displaystyle a{x}^3+b{x}^2+cx+d}$ is zero, then

• either, if ${\displaystyle b^2=3ac,}$ the cubic has a triple root $${\displaystyle x_1=x_2=x_3=-\frac{b}{3a},}$$ and $${\displaystyle a{x}^3+b{x}^2+cx+d=a{\left(x+\frac{b}{3a}\right)}^3}$$
• or, if ${\displaystyle b^2\ne 3ac,}$ the cubic has a double root $${\displaystyle x_2=x_3=\frac{9ad-bc}{2(b^2-3ac)},}$$ and a simple root, $${\displaystyle x_1=\frac{4abc-9a^{2}d-b^3}{a(b^2-3ac)}.}$$ and thus $${\displaystyle a{x}^3+b{x}^2+cx+d=a(x-x_1)(x-x_2{)}^2.}$$

Characteristic 2 and 3

The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.

The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its formal derivative. In these characteristics, if the derivative is not a constant, it is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas.

A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root.

Cardano's formula

Gerolamo Cardano is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro and Niccolo Fontana Tartaglia. The formula applies to depressed cubics, but, as shown in § Depressed cubic, it allows solving all cubic equations.

Cardano's result is that if $${\displaystyle t^3+pt+q=0}$$ is a cubic equation such that p and q are real numbers such that ${\displaystyle \frac{q^2}{4}+\frac{p^3}{27}}$ is positive (this implies that the discriminant of the equation is negative) then the equation has the real root ${\displaystyle \sqrt[3]{u_1}+\sqrt[3]{u_2},}$ where ${\displaystyle u_1}$ and ${\displaystyle u_2}$ are the two numbers ${\displaystyle -\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$ and ${\displaystyle -\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$.

See § Derivation of the roots, below, for several methods for getting this result.

As shown in § Nature of the roots, the two other roots are non-real complex conjugate numbers, in this case. It was later shown (Cardano did not know complex numbers) that the two other roots are obtained by multiplying one of the cube roots by the primitive cube root of unity ${\displaystyle {\epsilon }_1=\frac{-1+i\sqrt{3}}{2},}$ and the other cube root by the other primitive cube root of the unity ${\displaystyle {\epsilon }_2={\epsilon }_1^2=\frac{-1-i\sqrt{3}}{2}.}$ That is, the other roots of the equation are ${\displaystyle {\epsilon }_1\sqrt[3]{u_1}+{\epsilon }_2\sqrt[3]{u_2}}$ and ${\displaystyle {\epsilon }_2\sqrt[3]{u_1}+{\epsilon }_1\sqrt[3]{u_2}.}$

If ${\displaystyle 4p^3+27q^2<0,}$ there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called casus irreducibilis, meaning irreducible case in Latin.

In casus irreducibilis, Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols ${\displaystyle \sqrt{{}^{}}}$ and ${\displaystyle \sqrt[3]{{}^{}}}$ as representing the principal values of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general.

The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be −p / 3. It results that a root of the equation is $${\displaystyle C-\frac{p}{3C}\text{with}C=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.}$$ In this formula, the symbols ${\displaystyle \sqrt{{}^{}}}$ and ${\displaystyle \sqrt[3]{{}^{}}}$ denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is ${\displaystyle \frac{-1\pm \sqrt{-3}}{2}.}$

This formula for the roots is always correct except when p = q = 0, with the proviso that if p = 0, the square root is chosen so that C ≠ 0. However, Cardano's formula is useless if ${\displaystyle p=0,}$ as the roots are the cube roots of ${\displaystyle -q.}$ Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not irreducible; this includes the case ${\displaystyle 4p^3+27q^2=0.}$

This formula is also correct when p and q belong to any field of characteristic other than 2 or 3.

General cubic formula

A cubic formula for the roots of the general cubic equation (with a ≠ 0) $${\displaystyle a{x}^3+b{x}^2+cx+d=0}$$ can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for complex coefficients, but also for coefficients a, b, c, d belonging to any algebraically closed field of characteristic other than 2 or 3. If the coefficients are real numbers, the formula covers all complex solutions, not just real ones.

The formula being rather complicated, it is worth splitting it in smaller formulas.

Let $${\displaystyle \begin{array}{rl}{\mathrm{\Delta }}_0& =b^2-3ac,\\ {\mathrm{\Delta }}_1& =2b^3-9abc+27a^{2}d.\end{array}}$$

(Both ${\displaystyle {\mathrm{\Delta }}_0}$ and ${\displaystyle {\mathrm{\Delta }}_1}$ can be expressed as resultants of the cubic and its derivatives: ${\displaystyle {\mathrm{\Delta }}_1}$ is ⁠−1/8a⁠ times the resultant of the cubic and its second derivative, and ${\displaystyle {\mathrm{\Delta }}_0}$ is ⁠−1/12a⁠ times the resultant of the first and second derivatives of the cubic polynomial.)

Then let $${\displaystyle C=\sqrt[3]{\frac{{\mathrm{\Delta }}_1\pm \sqrt{{\mathrm{\Delta }}_1^2-4{\mathrm{\Delta }}_0^3}}{2}},}$$ where the symbols ${\displaystyle \sqrt{{}^{}}}$ and ${\displaystyle \sqrt[3]{{}^{}}}$ are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "±" before the square root is either "+" or "–"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields C = 0 (this occurs if ${\displaystyle {\mathrm{\Delta }}_0=0}$), then the other sign must be selected instead. If both choices yield C = 0, that is, if ${\displaystyle {\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=0,}$ a fraction ⁠0/0⁠ occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section). With these conventions, one of the roots is $${\displaystyle x=-\frac{1}{3a}\left(b+C+\frac{{\mathrm{\Delta }}_0}{C}\right).}$$

The other two roots can be obtained by changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a primitive cube root of unity, that is ⁠–1 ± √–3/2⁠. In other words, the three roots are $${\displaystyle x_k=-\frac{1}{3a}\left(b+{\xi }^{k}C+\frac{{\mathrm{\Delta }}_0}{{\xi }^{k}C}\right),k\in \{0,1,2\}\text{,}}$$ where ξ = ⁠–1 + √–3/2⁠.

As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if ${\displaystyle {\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=0,}$ the formula gives that the three roots equal ${\displaystyle \frac{-b}{3a},}$ which means that the cubic polynomial can be factored as ${\displaystyle a(x+\frac{b}{3a}{)}^3.}$ A straightforward computation allows verifying that the existence of this factorization is equivalent with ${\displaystyle {\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=0.}$

Trigonometric and hyperbolic solutions

Trigonometric solution for three real roots

When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. Galois theory allows proving that when the three roots are real, and none is rational (casus irreducibilis), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines. More precisely, the roots of the depressed cubic $${\displaystyle t^3+pt+q=0}$$ are $${\displaystyle t_k=2\sqrt{-\frac{p}{3}}\cos \left[\frac{1}{3}\arccos \left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi k}{3}\right]\text{for }k=0,1,2.}$$

This formula is due to François Viète. It is purely real when the equation has three real roots (that is ${\displaystyle 4p^3+27q^2<0}$). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when p = 0.

This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic.

The formula can be proved as follows: Starting from the equation t³ + pt + q = 0, let us set  t = u cos θ. The idea is to choose u to make the equation coincide with the identity $${\displaystyle 4{\cos }^3\theta -3\cos \theta -\cos (3\theta )=0.}$$ For this, choose ${\displaystyle u=2\sqrt{-\frac{p}{3}},}$ and divide the equation by ${\displaystyle \frac{u^3}{4}.}$ This gives $${\displaystyle 4{\cos }^3\theta -3\cos \theta -\frac{3q}{2p}\sqrt{\frac{-3}{p}}=0.}$$ Combining with the above identity, one gets $${\displaystyle \cos (3\theta )=\frac{3q}{2p}\sqrt{\frac{-3}{p}},}$$ and the roots are thus $${\displaystyle t_k=2\sqrt{-\frac{p}{3}}\cos \left[\frac{1}{3}\arccos \left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-\frac{2\pi k}{3}\right]\text{for }k=0,1,2.}$$

Hyperbolic solution for one real root

When there is only one real root (and p ≠ 0), this root can be similarly represented using hyperbolic functions, as$${\displaystyle \begin{array}{rl}{t}_0& =-2\frac{|q|}{q}\sqrt{-\frac{p}{3}}\cosh \left[\frac{1}{3}\mathrm{arcosh}\left(\frac{-3|q|}{2p}\sqrt{\frac{-3}{p}}\right)\right]\text{if }4p^3+27q^2>0\text{ and }p<0,\\ t_0& =-2\sqrt{\frac{p}{3}}\sinh \left[\frac{1}{3}\mathrm{arsinh}\left(\frac{3q}{2p}\sqrt{\frac{3}{p}}\right)\right]\text{if }p>0.\end{array}}$$ If p ≠ 0 and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.

When p = ±3, the above values of t₀ are sometimes called the Chebyshev cube root. More precisely, the values involving cosines and hyperbolic cosines define, when p = −3, the same analytic function denoted C_1/3(q), which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S_1/3(q), when p = 3.

Geometric solutions

Omar Khayyám's solution

Omar Khayyám's geometric solution of a cubic equation, for the case m = 2, n = 16, giving the root 2. The intersection of the vertical line on the x-axis at the center of the circle is happenstance of the example illustrated.

For solving the cubic equation x³ + m²x = n where n > 0, Omar Khayyám constructed the parabola y = x²/m, the circle that has as a diameter the line segment [0, n/m²] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis (see the figure).

A simple modern proof is as follows. Multiplying the equation by x/m² and regrouping the terms gives $${\displaystyle \frac{x^4}{m^2}=x\left(\frac{n}{m^2}-x\right).}$$ The left-hand side is the value of y² on the parabola. The equation of the circle being y² + x(x − ⁠n/m²⁠) = 0, the right hand side is the value of y² on the circle.

Solution with angle trisector

A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots.

A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.

Geometric interpretation of the roots

Three real roots

For the cubic (1) with three real roots, the roots are the projection on the x-axis of the vertices A, B, and C of an equilateral triangle. The center of the triangle has the same x-coordinate as the inflection point.

Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form (2), t³ + pt + q = 0, as shown above, the solution can be expressed as

$${\displaystyle t_k=2\sqrt{-\frac{p}{3}}\cos \left(\frac{1}{3}\arccos \left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi }{3}\right)\text{for}k=0,1,2.}$$

Here ${\displaystyle \arccos \left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)}$ is an angle in the unit circle; taking ⁠1/3⁠ of that angle corresponds to taking a cube root of a complex number; adding −k⁠2π/3⁠ for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by ${\displaystyle 2\sqrt{-\frac{p}{3}}}$ corrects for scale.

For the non-depressed case (1) (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that x = t − ⁠b/3a⁠ so t = x + ⁠b/3a⁠. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the y-axis. Consequently, the roots of the equation in t sum to zero.

One real root

In the Cartesian plane

The slope of line RA is twice that of RH. Denoting the complex roots of the cubic as g ± hi, g = OM (negative here) and h = √tan ORH = √slope of line RH = BE = DA.

When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa (x-coordinate) of the horizontal intercept of the curve (point R on the figure). Further, if the complex conjugate roots are written as g ± hi, then the real part g is the abscissa of the tangency point H of the tangent line to cubic that passes through x-intercept R of the cubic (that is the signed length OM, negative on the figure). The imaginary parts ±h are the square roots of the tangent of the angle between this tangent line and the horizontal axis.

In the complex plane

With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.

The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than ⁠π/3⁠ then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than ⁠π/3⁠, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is ⁠π/3⁠, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.

Galois group

Given a cubic irreducible polynomial over a field K of characteristic different from 2 and 3, the Galois group over K is the group of the field automorphisms that fix K of the smallest extension of K (splitting field). As these automorphisms must permute the roots of the polynomials, this group is either the group S₃ of all six permutations of the three roots, or the group A₃ of the three circular permutations.

The discriminant Δ of the cubic is the square of $${\displaystyle \sqrt{\mathrm{\Delta }}=a^2(r_1-r_2)(r_1-r_3)(r_2-r_3),}$$ where a is the leading coefficient of the cubic, and r₁, r₂ and r₃ are the three roots of the cubic. As ${\displaystyle \sqrt{\mathrm{\Delta }}}$ changes sign if two roots are exchanged, ${\displaystyle \sqrt{\mathrm{\Delta }}}$ is fixed by the Galois group only if the Galois group is A₃. In other words, the Galois group is A₃ if and only if the discriminant is the square of an element of K.

As most integers are not squares, when working over the field Q of the rational numbers, the Galois group of most irreducible cubic polynomials is the group S₃ with six elements. An example of a Galois group A₃ with three elements is given by p(x) = x³ − 3x − 1, whose discriminant is 81 = 9².

Derivation of the roots

This section regroups several methods for deriving Cardano's formula.

Cardano's method

This method is due to Scipione del Ferro and Tartaglia, but is named after Gerolamo Cardano who first published it in his book Ars Magna (1545).

This method applies to a depressed cubic t³ + pt + q = 0. The idea is to introduce two variables u and ${\displaystyle v}$ such that ${\displaystyle u+v=t}$ and to substitute this in the depressed cubic, giving $${\displaystyle u^3+v^3+(3uv+p)(u+v)+q=0.}$$

At this point Cardano imposed the condition ${\displaystyle 3uv+p=0.}$ This removes the third term in the previous equality, leading to the system of equations $${\displaystyle \begin{array}{rl}{u}^3+v^3& =-q\\ uv& =-\frac{p}{3}.\end{array}}$$

Knowing the sum and the product of u³ and ${\displaystyle v^3,}$ one deduces that they are the two solutions of the quadratic equation $${\displaystyle \begin{array}{rl}0& =(x-u^3)(x-v^3)\\ & =x^2-(u^3+v^3)x+u^3{v}^3\\ & =x^2-(u^3+v^3)x+(uv{)}^3\end{array}}$$ so $${\displaystyle x^2+qx-\frac{p^3}{27}=0.}$$ The discriminant of this equation is ${\displaystyle \mathrm{\Delta }=q^2+\frac{4p^3}{27}}$, and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root): $${\displaystyle -\frac{q}{2}\pm \sqrt{\frac{q^2}{4}+\frac{p^3}{27}}.}$$ So (without loss of generality in choosing u or ${\displaystyle v}$): $${\displaystyle u=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.}$$ $${\displaystyle v=\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.}$$ As ${\displaystyle u+v=t,}$ the sum of the cube roots of these solutions is a root of the equation. That is $${\displaystyle t=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}}$$ is a root of the equation; this is Cardano's formula.

This works well when ${\displaystyle 4p^3+27q^2>0,}$ but, if ${\displaystyle 4p^3+27q^2<0,}$ the square root appearing in the formula is not real. As a complex number has three cube roots, using Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by Rafael Bombelli in his book L'Algebra (1572). The solution is to use the fact that ${\displaystyle uv=-\frac{p}{3},}$ that is, ${\displaystyle v=\frac{-p}{3u}.}$ This means that only one cube root needs to be computed, and leads to the second formula given in § Cardano's formula.

The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two primitive cube roots of unity, which are ${\displaystyle \frac{-1\pm \sqrt{-3}}{2}.}$

When only one root is real, u and v will be the complex conjugates of one another, implying that the one real root must be ${\displaystyle t=2\mathrm{R}\mathrm{e}(u)}$.

Vieta's substitution

Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots.

Starting from the depressed cubic t³ + pt + q = 0, Vieta's substitution is t = w − ⁠p/3w⁠.

The substitution t = w – ⁠p/3w⁠ transforms the depressed cubic into $${\displaystyle w^3+q-\frac{p^3}{27w^3}=0.}$$

This is a quadratic equation in ${\displaystyle w^3}$, so there are six solutions for ${\displaystyle w}$. In the substitution, for each value of ${\displaystyle t}$ there are two possible values for ${\displaystyle w}$. Each root of the cubic equation is found twice.

Multiplying by w³, one gets a quadratic equation in w³: $${\displaystyle (w^3{)}^2+q(w^3)-\frac{p^3}{27}=0.}$$

Let $${\displaystyle W=-\frac{q}{2}\pm \sqrt{\frac{p^3}{27}+\frac{q^2}{4}}}$$ be any nonzero root of this quadratic equation. If w₁, w₂ and w₃ are the three cube roots of W, then the roots of the original depressed cubic are w₁ − ⁠p/3w₁⁠, w₂ − ⁠p/3w₂⁠, and w₃ − ⁠p/3w₃⁠. The other root of the quadratic equation is ${\displaystyle -\frac{p^3}{27W}.}$ This implies that changing the sign of the square root exchanges w_i and − ⁠p/3w_i⁠ for i = 1, 2, 3, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when p = q = 0, in which case the only root of the depressed cubic is 0.

Lagrange's method

In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees. This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six. Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher; as was later proved by the Abel–Ruffini theorem. Nevertheless, modern methods for solving solvable quintic equations are mainly based on Lagrange's method.

In the case of cubic equations, Lagrange's method gives the same solution as Cardano's. Lagrange's method can be applied directly to the general cubic equation ax³ + bx² + cx + d = 0, but the computation is simpler with the depressed cubic equation, t³ + pt + q = 0.

Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves. More precisely, let ξ be a primitive third root of unity, that is a number such that ξ³ = 1 and ξ² + ξ + 1 = 0 (when working in the space of complex numbers, one has ${\displaystyle \xi =\frac{-1\pm i\sqrt{3}}{2}=e^{2i\pi /3},}$ but this complex interpretation is not used here). Denoting x₀, x₁ and x₂ the three roots of the cubic equation to be solved, let $${\displaystyle \begin{array}{rl}{s}_0& =x_0+x_1+x_2,\\ s_1& =x_0+\xi x_1+{\xi }^2{x}_2,\\ s_2& =x_0+{\xi }^2{x}_1+\xi x_2,\end{array}}$$ be the discrete Fourier transform of the roots. If s₀, s₁ and s₂ are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is, $${\displaystyle \begin{array}{rl}{x}_0& =\frac{1}{3}(s_0+s_1+s_2),\\ x_1& =\frac{1}{3}(s_0+{\xi }^2{s}_1+\xi s_2),\\ x_2& =\frac{1}{3}(s_0+\xi s_1+{\xi }^2{s}_2).\end{array}}$$

By Vieta's formulas, s₀ is known to be zero in the case of a depressed cubic, and −⁠b/a⁠ for the general cubic. So, only s₁ and s₂ need to be computed. They are not symmetric functions of the roots (exchanging x₁ and x₂ exchanges also s₁ and s₂), but some simple symmetric functions of s₁ and s₂ are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the s_i as roots of a polynomial with known coefficients. This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the s_i as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher.

In the case of a cubic equation, ${\displaystyle P=s_1{s}_2,}$ and ${\displaystyle S=s_1^3+s_2^3}$ are such symmetric polynomials (see below). It follows that ${\displaystyle s_1^3}$ and ${\displaystyle s_2^3}$ are the two roots of the quadratic equation ${\displaystyle z^2-Sz+P^3=0.}$ Thus the resolution of the equation may be finished exactly as with Cardano's method, with ${\displaystyle s_1}$ and ${\displaystyle s_2}$ in place of u and ${\displaystyle v.}$

In the case of the depressed cubic, one has ${\displaystyle x_0=\frac{1}{3}(s_1+s_2)}$ and ${\displaystyle s_1{s}_2=-3p,}$ while in Cardano's method we have set ${\displaystyle x_0=u+v}$ and ${\displaystyle uv=-\frac{1}{3}p.}$ Thus, up to the exchange of u and ${\displaystyle v,}$ we have ${\displaystyle s_1=3u}$ and ${\displaystyle s_2=3v.}$ In other words, in this case, Cardano's method and Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem.

Computation of S and P

A straightforward computation using the relations ξ³ = 1 and ξ² + ξ + 1 = 0 gives $${\displaystyle \begin{array}{rl}P& =s_1{s}_2=x_0^2+x_1^2+x_2^2-(x_0{x}_1+x_1{x}_2+x_2{x}_0),\\ S& =s_1^3+s_2^3=2(x_0^3+x_1^3+x_2^3)-3(x_0^2{x}_1+x_1^2{x}_2+x_2^2{x}_0+x_0{x}_1^2+x_1{x}_2^2+x_2{x}_0^2)+12x_0{x}_1{x}_2.\end{array}}$$ This shows that P and S are symmetric functions of the roots. Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving $${\displaystyle \begin{array}{rl}P& =e_1^2-3e_2,\\ S& =2e_1^3-9e_1{e}_2+27e_3,\end{array}}$$ with e₁ = 0, e₂ = p and e₃ = −q in the case of a depressed cubic, and e₁ = −⁠b/a⁠, e₂ = ⁠c/a⁠ and e₃ = −⁠d/a⁠, in the general case.

Applications

Cubic equations arise in various other contexts.

In mathematics

• Angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.
• Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.
• The area of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to ${\displaystyle 2\pi /7}$ satisfy cubic equations.
• Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.
• The solution of the general quartic equation relies on the solution of its resolvent cubic.
• The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix.
• The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation.
• Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve.
• Critical points of a quartic function are found by solving a cubic equation (the derivative set equal to zero).
• Inflection points of a quintic function are the solution of a cubic equation (the second derivative set equal to zero).

In other sciences

• In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation.
• In thermodynamics, equations of state (which relate pressure, volume, and temperature of a substances), e.g. the Van der Waals equation of state, are cubic in the volume.
• Kinematic equations involving linear rates of acceleration are cubic.
• The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation.
• The steady state speed of a vehicle moving on a slope with air friction for a given input power is solved by a depressed cubic equation.
• Kepler's third law of planetary motion is cubic in the semi-major axis.